1. A(n) _____ _______ is a function that associates a vector to a point in the plane or to a point in space.

2. True or False Let $P=P(x,y,z)$, $Q=Q(x,y,z)$, and $R=R(x,y,z)$ be functions of three variables defined on a subset $E$ of space. A vector field over $E$ is defined as the function $F(x,y,z)=P(x,y,z)+Q(x,y,z)+R(x,y,z)$.

3. The domain of the vector field $\mathbf{F}=\mathbf{F}(x,y)$ is a set of points $(x,y)$ in the plane, and the range of $\mathbf{F}$ is a set of ______ in the plane.

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4. True or False The gradient of a function $f$ is an example of a vector field.

5. $\mathbf{F}=\mathbf{F}(x,y)=x\mathbf{i}+y\mathbf{j}$

The vector field is pictured below. See the Student Solutions Manual for a description.

6. $\mathbf{F}=\mathbf{F}(x,y)=x\mathbf{i}-y\mathbf{j}$

7. $\mathbf{F}=\mathbf{F}(x,y)=\mathbf{i}+x\mathbf{j}$

The vector field is pictured below. See the Student Solutions Manual for a description.

8. $\mathbf{F}=\mathbf{F}(x,y)=y\mathbf{i}-\mathbf{j}$

9. $\mathbf{F}=\mathbf{F}(x,y)=\mathbf{i}$

The vector field is pictured below. See the Student Solutions Manual for a description.

10. $\mathbf{F}=\mathbf{F}(x,y)=-\mathbf{j}$

11. $\mathbf{F}=\mathbf{F}(x,y)=\mathbf{i}+\mathbf{j}$

The vector field is pictured below. See the Student Solutions Manual for a description.

12. $\mathbf{F}=\mathbf{F}(x,y)=-\mathbf{i}+\mathbf{j}$

13. $\mathbf{F}=\mathbf{F}(x,y,z)=z \mathbf{k}$

The vector field is pictured below. See the Student Solutions Manual for a description.

14. $\mathbf{F}=\mathbf{F}(x,y,z)=x\mathbf{i}$

15. $\mathbf{F}=\mathbf{F}(x,y,z)=\dfrac{x\mathbf{i}+y\mathbf{j}+z\mathbf{k}}{\sqrt{x^{2}+y^{2}+z^{2}}}$

The vector field is pictured below. See the Student Solutions Manual for a description.

16. $\mathbf{F}=\mathbf{F}(x,y,z)=-\dfrac{x\mathbf{i}+y\mathbf{j}+z\mathbf{k}}{\sqrt{x^{2}+y^{2}+z^{2}}}$

17. $f(x,y) =x\sin y+\cos y$

18. $f(x,y) =xe^{y}$

19. $f( x,y,z) =x^{2}y+xy+y^{2}z$

20. $f(x,y,z) =x^{2}y+xyz^{2}$

21. 1. (a) Show that the vector field from Example 1, $\mathbf{F}=\mathbf{F}(x,y)=-y\mathbf{i}+x\mathbf{j}$, is a set of vectors tangent to circles centered at the origin.
    2. (b) Confirm that the magnitude of each vector equals the radius of the circle.

22. Gravitational Potential Energy  The gravitational field $\bf{g}$ due to a very small object of mass $m {\rm kg}$ that is $r$ meters (m) from a large object is given by $\mathbf{g}=\!-\dfrac{Gm}{r^{3}}\mathbf{r}$, where $G=6.67\times\! 10^{-11}{\rm N}{\rm m}^{2}\!/\!{\rm kg}^{2}$ and $\mathbf{r}={x}\mathbf{i}+y\mathbf{j}+z\mathbf{k}$. Show that the gravitational field is a gradient vector field. That is, show that $\mathbf{g}=- {\boldsymbol\nabla }u,$ where $u=-\dfrac{Gm}{r}$. The scalar function $u$ is called the gravitational potential due to the mass $m$.